Random Number Generator - arc4Random - Numbers can not be same - swift2

I have a random number generator using arc4Random and integers with three numbers, number1, number2, number3, in the range 1 - 3. A zero is not permitted. That part I have sorted due to a previous request on here, however, when the test button is pressed, I want to ensure the random numbers are not the same, i.e. if number1 displays 2, then number2 and number3 can not display 2 etc.
I tried to do this longhand below, but there must be a simpler way of doing it.
How would this be written in code as I have only been learning Xcode for about 6 weeks?
Regards.
Del Hinds
#IBAction func testButtonPressed(sender: UIButton) {
var str = "Hello, playground"
var number1 = Int(arc4random() % UInt32(4))
if number1 == 0 {
number1 = 1
}
var number2 = Int(arc4random() % UInt32(4))
if number2 == 0 {
if number1 == 1 {
number2 = 2
}
else if number1 == 2 {
number2 = 3
}
else if number1 == 3 {
number2 = 1
}
number2 = 1
}
label2TextLabel.text = "\(number2)"
var number3 = Int(arc4random() % UInt32(4))
if number3 == 0 {
if number2 == 1 {
number3 = 2
}
else if number2 == 2 {
number3 = 3
}
else if number2 == 3 {
number3 = 1
}
number3 = 1
}
label3TextLabel.text = "\(number3)"

If you are using ios9's GameplayKit, consider GKShuffledDistribution.
Playground example:
import GameplayKit
let distribution = GKShuffledDistribution(lowestValue: 1, highestValue: 3)
for i in 1...36 { // Do 12 groups of 3
print(distribution.nextInt(), appendnewline: false)
if i % 3 == 0 { print(" ", appendnewline: false ) }
}
Sample result:
213 132 321 312 213 132 132 123 132 231 132 123
Or consider GKRandomSource:
import GameplayKit
blocks : [AnyObject] = ["A","B","C"]
for _ in 1...5 { // do 5 shuffles
blocks = GKRandomSource.sharedRandom().arrayByShufflingObjectsInArray(blocks)
print(blocks)
}
Sample result:
[A, C, B]
[B, C, A]
[B, A, C]
[C, A, B]
[B, A, C]
Swift 3:
GKShuffledDistribution - No real change, modified print usage.
import GameplayKit
let distribution = GKShuffledDistribution(lowestValue: 1, highestValue: 3)
for i in 1...36 {
print(distribution.nextInt(), terminator: "")
if i % 3 == 0 { print(" ", terminator: "" ) }
}
GKRandomSource usage revision
Replaced AnyObject with Any
Now use arrayByShufflingObjects(in: blocks)
gives
import GameplayKit
blocks : [Any] = ["A","B","C"]
for _ in 1...5 { // do 5 shuffles
blocks = GKRandomSource.sharedRandom().arrayByShufflingObjects(in: blocks)
print(blocks)
}

A crude way of doing this is to start with an array of the possible number set {1, 2, 3} in your case, and, swap random elements in the array a few times, using the built-in random generator.
If however you require the shuffling to have the statistical property that the probability of a given number occurring conditional on it having not already occurred is 1 / (numbers remaining), then adopt the more rigorous Fisher-Yates shuffle.

Create an array (or set) of assigned numbers. When assigning a new number, check to see if the number is already in the array. If it is then assign a new random number. Repeat.
Once a number is assigned, add it to the array. Check if the array size is greater than or equal to the range of permitted numbers. If the number of random numbers is exhausted then throw an error and don't allow more numbers to be randomly assigned.
If you're doing this for large sets of numbers, you should check to make sure the size of the array of used numbers is not larger than about half the size of the range of assignable numbers.
If you're doing this for a small set of numbers, then it would be better to add the numbers in the range to an array and remove them from the array as you assign them.

Related

ELM recursive iteration

I need to create a function that takes a number, such as for a given number x, it computes the number y, by adding all digits of number x to itself.
An example :
Given x = 123:
It return 129 = 1 + 2 + 3 + 123
Given x = 35:
It return y = 43 = 3 + 5 + 35
I have this function that works but I need another way:
computeNextValue : Int -> Int
computeNextValue input =
String.fromInt input
|> String.split ""
|> List.filterMap String.toInt
|> List.sum
|> (+) input
First, let's make a recursive function that gives you all the digits of a number. The main idea is that you can get the rightmost digit of an integer by modBy 10, and you can remove it from the number by // 10:
getDigits : Int -> List Int
getDigits num =
if num == 0 then
[] -- base case
else
modBy 10 num :: getDigits (num // 10) -- recursive case
Note that this function returns the [] for 0, but that's OK for this usecase.
computeNextValue : Int -> Int
computeNextValue input =
input + List.sum (getDigits inputs)

Algorithm for equiprobable random square binary matrices with two non-adjacent non-zeros in each row and column

It would be great if someone could point me towards an algorithm that would allow me to :
create a random square matrix, with entries 0 and 1, such that
every row and every column contain exactly two non-zero entries,
two non-zero entries cannot be adjacent,
all possible matrices are equiprobable.
Right now I manage to achieve points 1 and 2 doing the following : such a matrix can be transformed, using suitable permutations of rows and columns, into a diagonal block matrix with blocks of the form
1 1 0 0 ... 0
0 1 1 0 ... 0
0 0 1 1 ... 0
.............
1 0 0 0 ... 1
So I start from such a matrix using a partition of [0, ..., n-1] and scramble it by permuting rows and columns randomly. Unfortunately, I can't find a way to integrate the adjacency condition, and I am quite sure that my algorithm won't treat all the matrices equally.
Update
I have managed to achieve point 3. The answer was actually straight under my nose : the block matrix I am creating contains all the information needed to take into account the adjacency condition. First some properties and definitions:
a suitable matrix defines permutations of [1, ..., n] that can be build like so: select a 1 in row 1. The column containing this entry contains exactly one other entry equal to 1 on a row a different from 1. Again, row a contains another entry 1 in a column which contains a second entry 1 on a row b, and so on. This starts a permutation 1 -> a -> b ....
For instance, with the following matrix, starting with the marked entry
v
1 0 1 0 0 0 | 1
0 1 0 0 0 1 | 2
1 0 0 1 0 0 | 3
0 0 1 0 1 0 | 4
0 0 0 1 0 1 | 5
0 1 0 0 1 0 | 6
------------+--
1 2 3 4 5 6 |
we get permutation 1 -> 3 -> 5 -> 2 -> 6 -> 4 -> 1.
the cycles of such a permutation lead to the block matrix I mentioned earlier. I also mentioned scrambling the block matrix using arbitrary permutations on the rows and columns to rebuild a matrix compatible with the requirements.
But I was using any permutation, which led to some adjacent non-zero entries. To avoid that, I have to choose permutations that separate rows (and columns) that are adjacent in the block matrix. Actually, to be more precise, if two rows belong to a same block and are cyclically consecutive (the first and last rows of a block are considered consecutive too), then the permutation I want to apply has to move these rows into non-consecutive rows of the final matrix (I will call two rows incompatible in that case).
So the question becomes : How to build all such permutations ?
The simplest idea is to build a permutation progressively by randomly adding rows that are compatible with the previous one. As an example, consider the case n = 6 using partition 6 = 3 + 3 and the corresponding block matrix
1 1 0 0 0 0 | 1
0 1 1 0 0 0 | 2
1 0 1 0 0 0 | 3
0 0 0 1 1 0 | 4
0 0 0 0 1 1 | 5
0 0 0 1 0 1 | 6
------------+--
1 2 3 4 5 6 |
Here rows 1, 2 and 3 are mutually incompatible, as are 4, 5 and 6. Choose a random row, say 3.
We will write a permutation as an array: [2, 5, 6, 4, 3, 1] meaning 1 -> 2, 2 -> 5, 3 -> 6, ... This means that row 2 of the block matrix will become the first row of the final matrix, row 5 will become the second row, and so on.
Now let's build a suitable permutation by choosing randomly a row, say 3:
p = [3, ...]
The next row will then be chosen randomly among the remaining rows that are compatible with 3 : 4, 5and 6. Say we choose 4:
p = [3, 4, ...]
Next choice has to be made among 1 and 2, for instance 1:
p = [3, 4, 1, ...]
And so on: p = [3, 4, 1, 5, 2, 6].
Applying this permutation to the block matrix, we get:
1 0 1 0 0 0 | 3
0 0 0 1 1 0 | 4
1 1 0 0 0 0 | 1
0 0 0 0 1 1 | 5
0 1 1 0 0 0 | 2
0 0 0 1 0 1 | 6
------------+--
1 2 3 4 5 6 |
Doing so, we manage to vertically isolate all non-zero entries. Same has to be done with the columns, for instance by using permutation p' = [6, 3, 5, 1, 4, 2] to finally get
0 1 0 1 0 0 | 3
0 0 1 0 1 0 | 4
0 0 0 1 0 1 | 1
1 0 1 0 0 0 | 5
0 1 0 0 0 1 | 2
1 0 0 0 1 0 | 6
------------+--
6 3 5 1 4 2 |
So this seems to work quite efficiently, but building these permutations needs to be done with caution, because one can easily be stuck: for instance, with n=6 and partition 6 = 2 + 2 + 2, following the construction rules set up earlier can lead to p = [1, 3, 2, 4, ...]. Unfortunately, 5 and 6 are incompatible, so choosing one or the other makes the last choice impossible. I think I've found all situations that lead to a dead end. I will denote by r the set of remaining choices:
p = [..., x, ?], r = {y} with x and y incompatible
p = [..., x, ?, ?], r = {y, z} with y and z being both incompatible with x (no choice can be made)
p = [..., ?, ?], r = {x, y} with x and y incompatible (any choice would lead to situation 1)
p = [..., ?, ?, ?], r = {x, y, z} with x, y and z being cyclically consecutive (choosing x or z would lead to situation 2, choosing y to situation 3)
p = [..., w, ?, ?, ?], r = {x, y, z} with xwy being a 3-cycle (neither x nor y can be chosen, choosing z would lead to situation 3)
p = [..., ?, ?, ?, ?], r = {w, x, y, z} with wxyz being a 4-cycle (any choice would lead to situation 4)
p = [..., ?, ?, ?, ?], r = {w, x, y, z} with xyz being a 3-cycle (choosing w would lead to situation 4, choosing any other would lead to situation 4)
Now it seems that the following algorithm gives all suitable permutations:
As long as there are strictly more than 5 numbers to choose, choose randomly among the compatible ones.
If there are 5 numbers left to choose: if the remaining numbers contain a 3-cycle or a 4-cycle, break that cycle (i.e. choose a number belonging to that cycle).
If there are 4 numbers left to choose: if the remaining numbers contain three cyclically consecutive numbers, choose one of them.
If there are 3 numbers left to choose: if the remaining numbers contain two cyclically consecutive numbers, choose one of them.
I am quite sure that this allows me to generate all suitable permutations and, hence, all suitable matrices.
Unfortunately, every matrix will be obtained several times, depending on the partition that was chosen.
Intro
Here is some prototype-approach, trying to solve the more general task of
uniform combinatorial sampling, which for our approach here means: we can use this approach for everything which we can formulate as SAT-problem.
It's not exploiting your problem directly and takes a heavy detour. This detour to the SAT-problem can help in regards to theory (more powerful general theoretical results) and efficiency (SAT-solvers).
That being said, it's not an approach if you want to sample within seconds or less (in my experiments), at least while being concerned about uniformity.
Theory
The approach, based on results from complexity-theory, follows this work:
GOMES, Carla P.; SABHARWAL, Ashish; SELMAN, Bart. Near-uniform sampling of combinatorial spaces using XOR constraints. In: Advances In Neural Information Processing Systems. 2007. S. 481-488.
The basic idea:
formulate the problem as SAT-problem
add randomly generated xors to the problem (acting on the decision-variables only! that's important in practice)
this will reduce the number of solutions (some solutions will get impossible)
do that in a loop (with tuned parameters) until only one solution is left!
search for some solution is being done by SAT-solvers or #SAT-solvers (=model-counting)
if there is more than one solution: no xors will be added but a complete restart will be done: add random-xors to the start-problem!
The guarantees:
when tuning the parameters right, this approach achieves near-uniform sampling
this tuning can be costly, as it's based on approximating the number of possible solutions
empirically this can also be costly!
Ante's answer, mentioning the number sequence A001499 actually gives a nice upper bound on the solution-space (as it's just ignoring adjacency-constraints!)
The drawbacks:
inefficient for large problems (in general; not necessarily compared to the alternatives like MCMC and co.)
need to change / reduce parameters to produce samples
those reduced parameters lose the theoretical guarantees
but empirically: good results are still possible!
Parameters:
In practice, the parameters are:
N: number of xors added
L: minimum number of variables part of one xor-constraint
U: maximum number of variables part of one xor-constraint
N is important to reduce the number of possible solutions. Given N constant, the other variables of course also have some effect on that.
Theory says (if i interpret correctly), that we should use L = R = 0.5 * #dec-vars.
This is impossible in practice here, as xor-constraints hurt SAT-solvers a lot!
Here some more scientific slides about the impact of L and U.
They call xors of size 8-20 short-XORS, while we will need to use even shorter ones later!
Implementation
Final version
Here is a pretty hacky implementation in python, using the XorSample scripts from here.
The underlying SAT-solver in use is Cryptominisat.
The code basically boils down to:
Transform the problem to conjunctive normal-form
as DIMACS-CNF
Implement the sampling-approach:
Calls XorSample (pipe-based + file-based)
Call SAT-solver (file-based)
Add samples to some file for later analysis
Code: (i hope i did warn you already about the code-quality)
from itertools import count
from time import time
import subprocess
import numpy as np
import os
import shelve
import uuid
import pickle
from random import SystemRandom
cryptogen = SystemRandom()
""" Helper functions """
# K-ARY CONSTRAINT GENERATION
# ###########################
# SINZ, Carsten. Towards an optimal CNF encoding of boolean cardinality constraints.
# CP, 2005, 3709. Jg., S. 827-831.
def next_var_index(start):
next_var = start
while(True):
yield next_var
next_var += 1
class s_index():
def __init__(self, start_index):
self.firstEnvVar = start_index
def next(self,i,j,k):
return self.firstEnvVar + i*k +j
def gen_seq_circuit(k, input_indices, next_var_index_gen):
cnf_string = ''
s_index_gen = s_index(next_var_index_gen.next())
# write clauses of first partial sum (i.e. i=0)
cnf_string += (str(-input_indices[0]) + ' ' + str(s_index_gen.next(0,0,k)) + ' 0\n')
for i in range(1, k):
cnf_string += (str(-s_index_gen.next(0, i, k)) + ' 0\n')
# write clauses for general case (i.e. 0 < i < n-1)
for i in range(1, len(input_indices)-1):
cnf_string += (str(-input_indices[i]) + ' ' + str(s_index_gen.next(i, 0, k)) + ' 0\n')
cnf_string += (str(-s_index_gen.next(i-1, 0, k)) + ' ' + str(s_index_gen.next(i, 0, k)) + ' 0\n')
for u in range(1, k):
cnf_string += (str(-input_indices[i]) + ' ' + str(-s_index_gen.next(i-1, u-1, k)) + ' ' + str(s_index_gen.next(i, u, k)) + ' 0\n')
cnf_string += (str(-s_index_gen.next(i-1, u, k)) + ' ' + str(s_index_gen.next(i, u, k)) + ' 0\n')
cnf_string += (str(-input_indices[i]) + ' ' + str(-s_index_gen.next(i-1, k-1, k)) + ' 0\n')
# last clause for last variable
cnf_string += (str(-input_indices[-1]) + ' ' + str(-s_index_gen.next(len(input_indices)-2, k-1, k)) + ' 0\n')
return (cnf_string, (len(input_indices)-1)*k, 2*len(input_indices)*k + len(input_indices) - 3*k - 1)
# K=2 clause GENERATION
# #####################
def gen_at_most_2_constraints(vars, start_var):
constraint_string = ''
used_clauses = 0
used_vars = 0
index_gen = next_var_index(start_var)
circuit = gen_seq_circuit(2, vars, index_gen)
constraint_string += circuit[0]
used_clauses += circuit[2]
used_vars += circuit[1]
start_var += circuit[1]
return [constraint_string, used_clauses, used_vars, start_var]
def gen_at_least_2_constraints(vars, start_var):
k = len(vars) - 2
vars = [-var for var in vars]
constraint_string = ''
used_clauses = 0
used_vars = 0
index_gen = next_var_index(start_var)
circuit = gen_seq_circuit(k, vars, index_gen)
constraint_string += circuit[0]
used_clauses += circuit[2]
used_vars += circuit[1]
start_var += circuit[1]
return [constraint_string, used_clauses, used_vars, start_var]
# Adjacency conflicts
# ###################
def get_all_adjacency_conflicts_4_neighborhood(N, X):
conflicts = set()
for x in range(N):
for y in range(N):
if x < (N-1):
conflicts.add(((x,y),(x+1,y)))
if y < (N-1):
conflicts.add(((x,y),(x,y+1)))
cnf = '' # slow string appends
for (var_a, var_b) in conflicts:
var_a_ = X[var_a]
var_b_ = X[var_b]
cnf += '-' + var_a_ + ' ' + '-' + var_b_ + ' 0 \n'
return cnf, len(conflicts)
# Build SAT-CNF
#############
def build_cnf(N, verbose=False):
var_counter = count(1)
N_CLAUSES = 0
X = np.zeros((N, N), dtype=object)
for a in range(N):
for b in range(N):
X[a,b] = str(next(var_counter))
# Adjacency constraints
CNF, N_CLAUSES = get_all_adjacency_conflicts_4_neighborhood(N, X)
# k=2 constraints
NEXT_VAR = N*N+1
for row in range(N):
constraint_string, used_clauses, used_vars, NEXT_VAR = gen_at_most_2_constraints(X[row, :].astype(int).tolist(), NEXT_VAR)
N_CLAUSES += used_clauses
CNF += constraint_string
constraint_string, used_clauses, used_vars, NEXT_VAR = gen_at_least_2_constraints(X[row, :].astype(int).tolist(), NEXT_VAR)
N_CLAUSES += used_clauses
CNF += constraint_string
for col in range(N):
constraint_string, used_clauses, used_vars, NEXT_VAR = gen_at_most_2_constraints(X[:, col].astype(int).tolist(), NEXT_VAR)
N_CLAUSES += used_clauses
CNF += constraint_string
constraint_string, used_clauses, used_vars, NEXT_VAR = gen_at_least_2_constraints(X[:, col].astype(int).tolist(), NEXT_VAR)
N_CLAUSES += used_clauses
CNF += constraint_string
# build final cnf
CNF = 'p cnf ' + str(NEXT_VAR-1) + ' ' + str(N_CLAUSES) + '\n' + CNF
return X, CNF, NEXT_VAR-1
# External tools
# ##############
def get_random_xor_problem(CNF_IN_fp, N_DEC_VARS, N_ALL_VARS, s, min_l, max_l):
# .cnf not part of arg!
p = subprocess.Popen(['./gen-wff', CNF_IN_fp,
str(N_DEC_VARS), str(N_ALL_VARS),
str(s), str(min_l), str(max_l), 'xored'],
stdin=subprocess.PIPE, stdout=subprocess.PIPE, stderr=subprocess.PIPE)
result = p.communicate()
os.remove(CNF_IN_fp + '-str-xored.xor') # file not needed
return CNF_IN_fp + '-str-xored.cnf'
def solve(CNF_IN_fp, N_DEC_VARS):
seed = cryptogen.randint(0, 2147483647) # actually no reason to do it; but can't hurt either
p = subprocess.Popen(["./cryptominisat5", '-t', '4', '-r', str(seed), CNF_IN_fp], stdin=subprocess.PIPE, stdout=subprocess.PIPE)
result = p.communicate()[0]
sat_line = result.find('s SATISFIABLE')
if sat_line != -1:
# solution found!
vars = parse_solution(result)[:N_DEC_VARS]
# forbid solution (DeMorgan)
negated_vars = list(map(lambda x: x*(-1), vars))
with open(CNF_IN_fp, 'a') as f:
f.write( (str(negated_vars)[1:-1] + ' 0\n').replace(',', ''))
# assume solve is treating last constraint despite not changing header!
# solve again
seed = cryptogen.randint(0, 2147483647)
p = subprocess.Popen(["./cryptominisat5", '-t', '4', '-r', str(seed), CNF_IN_fp], stdin=subprocess.PIPE, stdout=subprocess.PIPE)
result = p.communicate()[0]
sat_line = result.find('s SATISFIABLE')
if sat_line != -1:
os.remove(CNF_IN_fp) # not needed anymore
return True, False, None
else:
return True, True, vars
else:
return False, False, None
def parse_solution(output):
# assumes there is one
vars = []
for line in output.split("\n"):
if line:
if line[0] == 'v':
line_vars = list(map(lambda x: int(x), line.split()[1:]))
vars.extend(line_vars)
return vars
# Core-algorithm
# ##############
def xorsample(X, CNF_IN_fp, N_DEC_VARS, N_VARS, s, min_l, max_l):
start_time = time()
while True:
# add s random XOR constraints to F
xored_cnf_fp = get_random_xor_problem(CNF_IN_fp, N_DEC_VARS, N_VARS, s, min_l, max_l)
state_lvl1, state_lvl2, var_sol = solve(xored_cnf_fp, N_DEC_VARS)
print('------------')
if state_lvl1 and state_lvl2:
print('FOUND')
d = shelve.open('N_15_70_4_6_TO_PLOT')
d[str(uuid.uuid4())] = (pickle.dumps(var_sol), time() - start_time)
d.close()
return True
else:
if state_lvl1:
print('sol not unique')
else:
print('no sol found')
print('------------')
""" Run """
N = 15
N_DEC_VARS = N*N
X, CNF, N_VARS = build_cnf(N)
with open('my_problem.cnf', 'w') as f:
f.write(CNF)
counter = 0
while True:
print('sample: ', counter)
xorsample(X, 'my_problem', N_DEC_VARS, N_VARS, 70, 4, 6)
counter += 1
Output will look like (removed some warnings):
------------
no sol found
------------
------------
no sol found
------------
------------
no sol found
------------
------------
sol not unique
------------
------------
FOUND
Core: CNF-formulation
We introduce one variable for every cell of the matrix. N=20 means 400 binary-variables.
Adjancency:
Precalculate all symmetry-reduced conflicts and add conflict-clauses.
Basic theory:
a -> !b
<->
!a v !b (propositional logic)
Row/Col-wise Cardinality:
This is tough to express in CNF and naive approaches need an exponential number
of constraints.
We use some adder-circuit based encoding (SINZ, Carsten. Towards an optimal CNF encoding of boolean cardinality constraints) which introduces new auxiliary-variables.
Remark:
sum(var_set) <= k
<->
sum(negated(var_set)) >= len(var_set) - k
These SAT-encodings can be put into exact model-counters (for small N; e.g. < 9). The number of solutions equals Ante's results, which is a strong indication for a correct transformation!
There are also interesting approximate model-counters (also heavily based on xor-constraints) like approxMC which shows one more thing we can do with the SAT-formulation. But in practice i have not been able to use these (approxMC = autoconf; no comment).
Other experiments
I did also build a version using pblib, to use more powerful cardinality-formulations
for the SAT-CNF formulation. I did not try to use the C++-based API, but only the reduced pbencoder, which automatically selects some best encoding, which was way worse than my encoding used above (which is best is still a research-problem; often even redundant-constraints can help).
Empirical analysis
For the sake of obtaining some sample-size (given my patience), i only computed samples for N=15. In this case we used:
N=70 xors
L,U = 4,6
I also computed some samples for N=20 with (100,3,6), but this takes a few mins and we reduced the lower bound!
Visualization
Here some animation (strengthening my love-hate relationship with matplotlib):
Edit: And a (reduced) comparison to brute-force uniform-sampling with N=5 (NXOR,L,U = 4, 10, 30):
(I have not yet decided on the addition of the plotting-code. It's as ugly as the above one and people might look too much into my statistical shambles; normalizations and co.)
Theory
Statistical analysis is probably hard to do as the underlying problem is of such combinatoric nature. It's even not entirely obvious how that final cell-PDF should look like. In the case of N=odd, it's probably non-uniform and looks like a chess-board (i did brute-force check N=5 to observe this).
One thing we can be sure about (imho): symmetry!
Given a cell-PDF matrix, we should expect, that the matrix is symmetric (A = A.T).
This is checked in the visualization and the euclidean-norm of differences over time is plotted.
We can do the same on some other observation: observed pairings.
For N=3, we can observe the following pairs:
0,1
0,2
1,2
Now we can do this per-row and per-column and should expect symmetry too!
Sadly, it's probably not easy to say something about the variance and therefore the needed samples to speak about confidence!
Observation
According to my simplified perception, current-samples and the cell-PDF look good, although convergence is not achieved yet (or we are far away from uniformity).
The more important aspect are probably the two norms, nicely decreasing towards 0.
(yes; one could tune some algorithm for that by transposing with prob=0.5; but this is not done here as it would defeat it's purpose).
Potential next steps
Tune parameters
Check out the approach using #SAT-solvers / Model-counters instead of SAT-solvers
Try different CNF-formulations, especially in regards to cardinality-encodings and xor-encodings
XorSample is by default using tseitin-like encoding to get around exponentially grow
for smaller xors (as used) it might be a good idea to use naive encoding (which propagates faster)
XorSample supports that in theory; but the script's work differently in practice
Cryptominisat is known for dedicated XOR-handling (as it was build for analyzing cryptography including many xors) and might gain something by naive encoding (as inferring xors from blown-up CNFs is much harder)
More statistical-analysis
Get rid of XorSample scripts (shell + perl...)
Summary
The approach is very general
This code produces feasible samples
It should be not hard to prove, that every feasible solution can be sampled
Others have proven theoretical guarantees for uniformity for some params
does not hold for our params
Others have empirically / theoretically analyzed smaller parameters (in use here)
(Updated test results, example run-through and code snippets below.)
You can use dynamic programming to calculate the number of solutions resulting from every state (in a much more efficient way than a brute-force algorithm), and use those (pre-calculated) values to create equiprobable random solutions.
Consider the example of a 7x7 matrix; at the start, the state is:
0,0,0,0,0,0,0
meaning that there are seven adjacent unused columns. After adding two ones to the first row, the state could be e.g.:
0,1,0,0,1,0,0
with two columns that now have a one in them. After adding ones to the second row, the state could be e.g.:
0,1,1,0,1,0,1
After three rows are filled, there is a possibility that a column will have its maximum of two ones; this effectively splits the matrix into two independent zones:
1,1,1,0,2,0,1 -> 1,1,1,0 + 0,1
These zones are independent in the sense that the no-adjacent-ones rule has no effect when adding ones to different zones, and the order of the zones has no effect on the number of solutions.
In order to use these states as signatures for types of solutions, we have to transform them into a canonical notation. First, we have to take into account the fact that columns with only 1 one in them may be unusable in the next row, because they contain a one in the current row. So instead of a binary notation, we have to use a ternary notation, e.g.:
2,1,1,0 + 0,1
where the 2 means that this column was used in the current row (and not that there are 2 ones in the column). At the next step, we should then convert the twos back into ones.
Additionally, we can also mirror the seperate groups to put them into their lexicographically smallest notation:
2,1,1,0 + 0,1 -> 0,1,1,2 + 0,1
Lastly, we sort the seperate groups from small to large, and then lexicographically, so that a state in a larger matrix may be e.g.:
0,0 + 0,1 + 0,0,2 + 0,1,0 + 0,1,0,1
Then, when calculating the number of solutions resulting from each state, we can use memoization using the canonical notation of each state as a key.
Creating a dictionary of the states and the number of solutions for each of them only needs to be done once, and a table for larger matrices can probably be used for smaller matrices too.
Practically, you'd generate a random number between 0 and the total number of solutions, and then for every row, you'd look at the different states you could create from the current state, look at the number of unique solutions each one would generate, and see which option leads to the solution that corresponds with your randomly generated number.
Note that every state and the corresponding key can only occur in a particular row, so you can store the keys in seperate dictionaries per row.
TEST RESULTS
A first test using unoptimized JavaScript gave very promising results. With dynamic programming, calculating the number of solutions for a 10x10 matrix now takes a second, where a brute-force algorithm took several hours (and this is the part of the algorithm that only needs to be done once). The size of the dictionary with the signatures and numbers of solutions grows with a diminishing factor approaching 2.5 for each step in size; the time to generate it grows with a factor of around 3.
These are the number of solutions, states, signatures (total size of the dictionaries), and maximum number of signatures per row (largest dictionary per row) that are created:
size unique solutions states signatures max/row
4x4 2 9 6 2
5x5 16 73 26 8
6x6 722 514 107 40
7x7 33,988 2,870 411 152
8x8 2,215,764 13,485 1,411 596
9x9 179,431,924 56,375 4,510 1,983
10x10 17,849,077,140 218,038 13,453 5,672
11x11 2,138,979,146,276 801,266 38,314 14,491
12x12 304,243,884,374,412 2,847,885 104,764 35,803
13x13 50,702,643,217,809,908 9,901,431 278,561 96,414
14x14 9,789,567,606,147,948,364 33,911,578 723,306 238,359
15x15 2,168,538,331,223,656,364,084 114,897,838 1,845,861 548,409
16x16 546,386,962,452,256,865,969,596 ... 4,952,501 1,444,487
17x17 155,420,047,516,794,379,573,558,433 12,837,870 3,754,040
18x18 48,614,566,676,379,251,956,711,945,475 31,452,747 8,992,972
19x19 17,139,174,923,928,277,182,879,888,254,495 74,818,773 20,929,008
20x20 6,688,262,914,418,168,812,086,412,204,858,650 175,678,000 50,094,203
(Additional results obtained with C++, using a simple 128-bit integer implementation. To count the states, the code had to be run using each state as a seperate signature, which I was unable to do for the largest sizes. )
EXAMPLE
The dictionary for a 5x5 matrix looks like this:
row 0: 00000 -> 16 row 3: 101 -> 0
1112 -> 1
row 1: 20002 -> 2 1121 -> 1
00202 -> 4 1+01 -> 0
02002 -> 2 11+12 -> 2
02020 -> 2 1+121 -> 1
0+1+1 -> 0
row 2: 10212 -> 1 1+112 -> 1
12012 -> 1
12021 -> 2 row 4: 0 -> 0
12102 -> 1 11 -> 0
21012 -> 0 12 -> 0
02121 -> 3 1+1 -> 1
01212 -> 1 1+2 -> 0
The total number of solutions is 16; if we randomly pick a number from 0 to 15, e.g. 13, we can find the corresponding (i.e. the 14th) solution like this:
state: 00000
options: 10100 10010 10001 01010 01001 00101
signature: 00202 02002 20002 02020 02002 00202
solutions: 4 2 2 2 2 4
This tells us that the 14th solution is the 2nd solution of option 00101. The next step is:
state: 00101
options: 10010 01010
signature: 12102 02121
solutions: 1 3
This tells us that the 2nd solution is the 1st solution of option 01010. The next step is:
state: 01111
options: 10100 10001 00101
signature: 11+12 1112 1+01
solutions: 2 1 0
This tells us that the 1st solution is the 1st solution of option 10100. The next step is:
state: 11211
options: 01010 01001
signature: 1+1 1+1
solutions: 1 1
This tells us that the 1st solutions is the 1st solution of option 01010. The last step is:
state: 12221
options: 10001
And the 5x5 matrix corresponding to randomly chosen number 13 is:
0 0 1 0 1
0 1 0 1 0
1 0 1 0 0
0 1 0 1 0
1 0 0 0 1
And here's a quick'n'dirty code example; run the snippet to generate the signature and solution count dictionary, and generate a random 10x10 matrix (it takes a second to generate the dictionary; once that is done, it generates random solutions in half a millisecond):
function signature(state, prev) {
var zones = [], zone = [];
for (var i = 0; i < state.length; i++) {
if (state[i] == 2) {
if (zone.length) zones.push(mirror(zone));
zone = [];
}
else if (prev[i]) zone.push(3);
else zone.push(state[i]);
}
if (zone.length) zones.push(mirror(zone));
zones.sort(function(a,b) {return a.length - b.length || a - b;});
return zones.length ? zones.join("2") : "2";
function mirror(zone) {
var ltr = zone.join('');
zone.reverse();
var rtl = zone.join('');
return (ltr < rtl) ? ltr : rtl;
}
}
function memoize(n) {
var memo = [], empty = [];
for (var i = 0; i <= n; i++) memo[i] = [];
for (var i = 0; i < n; i++) empty[i] = 0;
memo[0][signature(empty, empty)] = next_row(empty, empty, 1);
return memo;
function next_row(state, prev, row) {
if (row > n) return 1;
var solutions = 0;
for (var i = 0; i < n - 2; i++) {
if (state[i] == 2 || prev[i] == 1) continue;
for (var j = i + 2; j < n; j++) {
if (state[j] == 2 || prev[j] == 1) continue;
var s = state.slice(), p = empty.slice();
++s[i]; ++s[j]; ++p[i]; ++p[j];
var sig = signature(s, p);
var sol = memo[row][sig];
if (sol == undefined)
memo[row][sig] = sol = next_row(s, p, row + 1);
solutions += sol;
}
}
return solutions;
}
}
function random_matrix(n, memo) {
var matrix = [], empty = [], state = [], prev = [];
for (var i = 0; i < n; i++) empty[i] = state[i] = prev[i] = 0;
var total = memo[0][signature(empty, empty)];
var pick = Math.floor(Math.random() * total);
document.write("solution " + pick.toLocaleString('en-US') +
" from a total of " + total.toLocaleString('en-US') + "<br>");
for (var row = 1; row <= n; row++) {
var options = find_options(state, prev);
for (var i in options) {
var state_copy = state.slice();
for (var j in state_copy) state_copy[j] += options[i][j];
var sig = signature(state_copy, options[i]);
var solutions = memo[row][sig];
if (pick < solutions) {
matrix.push(options[i].slice());
prev = options[i].slice();
state = state_copy.slice();
break;
}
else pick -= solutions;
}
}
return matrix;
function find_options(state, prev) {
var options = [];
for (var i = 0; i < n - 2; i++) {
if (state[i] == 2 || prev[i] == 1) continue;
for (var j = i + 2; j < n; j++) {
if (state[j] == 2 || prev[j] == 1) continue;
var option = empty.slice();
++option[i]; ++option[j];
options.push(option);
}
}
return options;
}
}
var size = 10;
var memo = memoize(size);
var matrix = random_matrix(size, memo);
for (var row in matrix) document.write(matrix[row] + "<br>");
The code snippet below shows the dictionary of signatures and solution counts for a matrix of size 10x10. I've used a slightly different signature format from the explanation above: the zones are delimited by a '2' instead of a plus sign, and a column which has a one in the previous row is marked with a '3' instead of a '2'. This shows how the keys could be stored in a file as integers with 2×N bits (padded with 2's).
function signature(state, prev) {
var zones = [], zone = [];
for (var i = 0; i < state.length; i++) {
if (state[i] == 2) {
if (zone.length) zones.push(mirror(zone));
zone = [];
}
else if (prev[i]) zone.push(3);
else zone.push(state[i]);
}
if (zone.length) zones.push(mirror(zone));
zones.sort(function(a,b) {return a.length - b.length || a - b;});
return zones.length ? zones.join("2") : "2";
function mirror(zone) {
var ltr = zone.join('');
zone.reverse();
var rtl = zone.join('');
return (ltr < rtl) ? ltr : rtl;
}
}
function memoize(n) {
var memo = [], empty = [];
for (var i = 0; i <= n; i++) memo[i] = [];
for (var i = 0; i < n; i++) empty[i] = 0;
memo[0][signature(empty, empty)] = next_row(empty, empty, 1);
return memo;
function next_row(state, prev, row) {
if (row > n) return 1;
var solutions = 0;
for (var i = 0; i < n - 2; i++) {
if (state[i] == 2 || prev[i] == 1) continue;
for (var j = i + 2; j < n; j++) {
if (state[j] == 2 || prev[j] == 1) continue;
var s = state.slice(), p = empty.slice();
++s[i]; ++s[j]; ++p[i]; ++p[j];
var sig = signature(s, p);
var sol = memo[row][sig];
if (sol == undefined)
memo[row][sig] = sol = next_row(s, p, row + 1);
solutions += sol;
}
}
return solutions;
}
}
var memo = memoize(10);
for (var i in memo) {
document.write("row " + i + ":<br>");
for (var j in memo[i]) {
document.write(""" + j + "": " + memo[i][j] + "<br>");
}
}
Just few thoughts. Number of matrices satisfying conditions for n <= 10:
3 0
4 2
5 16
6 722
7 33988
8 2215764
9 179431924
10 17849077140
Unfortunatelly there is no sequence with these numbers in OEIS.
There is one similar (A001499), without condition for neighbouring one's. Number of nxn matrices in this case is 'of order' as A001499's number of (n-1)x(n-1) matrices. That is to be expected since number
of ways to fill one row in this case, position 2 one's in n places with at least one zero between them is ((n-1) choose 2). Same as to position 2 one's in (n-1) places without the restriction.
I don't think there is an easy connection between these matrix of order n and A001499 matrix of order n-1, meaning that if we have A001499 matrix than we can construct some of these matrices.
With this, for n=20, number of matrices is >10^30. Quite a lot :-/
This solution use recursion in order to set the cell of the matrix one by one. If the random walk finish with an impossible solution then we rollback one step in the tree and we continue the random walk.
The algorithm is efficient and i think that the generated data are highly equiprobable.
package rndsqmatrix;
import java.util.ArrayList;
import java.util.Collections;
import java.util.List;
import java.util.stream.IntStream;
public class RndSqMatrix {
/**
* Generate a random matrix
* #param size the size of the matrix
* #return the matrix encoded in 1d array i=(x+y*size)
*/
public static int[] generate(final int size) {
return generate(size, new int[size * size], new int[size],
new int[size]);
}
/**
* Build a matrix recursivly with a random walk
* #param size the size of the matrix
* #param matrix the matrix encoded in 1d array i=(x+y*size)
* #param rowSum
* #param colSum
* #return
*/
private static int[] generate(final int size, final int[] matrix,
final int[] rowSum, final int[] colSum) {
// generate list of valid positions
final List<Integer> positions = new ArrayList();
for (int y = 0; y < size; y++) {
if (rowSum[y] < 2) {
for (int x = 0; x < size; x++) {
if (colSum[x] < 2) {
final int p = x + y * size;
if (matrix[p] == 0
&& (x == 0 || matrix[p - 1] == 0)
&& (x == size - 1 || matrix[p + 1] == 0)
&& (y == 0 || matrix[p - size] == 0)
&& (y == size - 1 || matrix[p + size] == 0)) {
positions.add(p);
}
}
}
}
}
// no valid positions ?
if (positions.isEmpty()) {
// if the matrix is incomplete => return null
for (int i = 0; i < size; i++) {
if (rowSum[i] != 2 || colSum[i] != 2) {
return null;
}
}
// the matrix is complete => return it
return matrix;
}
// random walk
Collections.shuffle(positions);
for (int p : positions) {
// set '1' and continue recursivly the exploration
matrix[p] = 1;
rowSum[p / size]++;
colSum[p % size]++;
final int[] solMatrix = generate(size, matrix, rowSum, colSum);
if (solMatrix != null) {
return solMatrix;
}
// rollback
matrix[p] = 0;
rowSum[p / size]--;
colSum[p % size]--;
}
// we can't find a valid matrix from here => return null
return null;
}
public static void printMatrix(final int size, final int[] matrix) {
for (int y = 0; y < size; y++) {
for (int x = 0; x < size; x++) {
System.out.print(matrix[x + y * size]);
System.out.print(" ");
}
System.out.println();
}
}
public static void printStatistics(final int size, final int count) {
final int sumMatrix[] = new int[size * size];
for (int i = 0; i < count; i++) {
final int[] matrix = generate(size);
for (int j = 0; j < sumMatrix.length; j++) {
sumMatrix[j] += matrix[j];
}
}
printMatrix(size, sumMatrix);
}
public static void checkAlgorithm() {
final int size = 8;
final int count = 2215764;
final int divisor = 122;
final int sumMatrix[] = new int[size * size];
for (int i = 0; i < count/divisor ; i++) {
final int[] matrix = generate(size);
for (int j = 0; j < sumMatrix.length; j++) {
sumMatrix[j] += matrix[j];
}
}
int total = 0;
for(int i=0; i < sumMatrix.length; i++) {
total += sumMatrix[i];
}
final double factor = (double)total / (count/divisor);
System.out.println("Factor=" + factor + " (theory=16.0)");
}
public static void benchmark(final int size, final int count,
final boolean parallel) {
final long begin = System.currentTimeMillis();
if (!parallel) {
for (int i = 0; i < count; i++) {
generate(size);
}
} else {
IntStream.range(0, count).parallel().forEach(i -> generate(size));
}
final long end = System.currentTimeMillis();
System.out.println("rate="
+ (double) (end - begin) / count + "ms/matrix");
}
public static void main(String[] args) {
checkAlgorithm();
benchmark(8, 10000, true);
//printStatistics(8, 2215764/36);
printStatistics(8, 2215764);
}
}
The output is:
Factor=16.0 (theory=16.0)
rate=0.2835ms/matrix
552969 554643 552895 554632 555680 552753 554567 553389
554071 554847 553441 553315 553425 553883 554485 554061
554272 552633 555130 553699 553604 554298 553864 554028
554118 554299 553565 552986 553786 554473 553530 554771
554474 553604 554473 554231 553617 553556 553581 553992
554960 554572 552861 552732 553782 554039 553921 554661
553578 553253 555721 554235 554107 553676 553776 553182
553086 553677 553442 555698 553527 554850 553804 553444
Here is a very fast approach of generating the matrix row by row, written in Java:
public static void main(String[] args) throws Exception {
int n = 100;
Random rnd = new Random();
byte[] mat = new byte[n*n];
byte[] colCount = new byte[n];
//generate row by row
for (int x = 0; x < n; x++) {
//generate a random first bit
int b1 = rnd.nextInt(n);
while ( (x > 0 && mat[(x-1)*n + b1] == 1) || //not adjacent to the one above
(colCount[b1] == 2) //not in a column which has 2
) b1 = rnd.nextInt(n);
//generate a second bit, not equal to the first one
int b2 = rnd.nextInt(n);
while ( (b2 == b1) || //not the same as bit 1
(x > 0 && mat[(x-1)*n + b2] == 1) || //not adjacent to the one above
(colCount[b2] == 2) || //not in a column which has 2
(b2 == b1 - 1) || //not adjacent to b1
(b2 == b1 + 1)
) b2 = rnd.nextInt(n);
//fill the matrix values and increment column counts
mat[x*n + b1] = 1;
mat[x*n + b2] = 1;
colCount[b1]++;
colCount[b2]++;
}
String arr = Arrays.toString(mat).substring(1, n*n*3 - 1);
System.out.println(arr.replaceAll("(.{" + n*3 + "})", "$1\n"));
}
It essentially generates each a random row at a time. If the row will violate any of the conditions, it is generated again (again randomly). I believe this will satisfy condition 4 as well.
Adding a quick note that it will spin forever for N-s where there is no solutions (like N=3).

Single Number 2 Scala solution

Given an array of integers, every element appears three times except for one, which appears exactly once. Find that single one. This is what I have now. But I don't know how to break the for loop once I get the single number "b". Any solution in scala please?
for(Array(a,b) <- nums.sorted.sliding(2))
{
if (a == b){j = j+1}
else
{
if (j < 3) j =1
b
}
}
This will do it.
nums.groupBy(identity).find(_._2.length == 1).get._1
It's a bit unsafe in that it will throw if there is no single-count element. It can be made safer if a default value is returned when no single-count element is found.
nums.groupBy(identity).find(_._2.length == 1).fold(-1)(_._1)
Another way is to sum the array by adding digits of two numbers in base 3 modulo 3 (in other words XOR in base 3). The elements that appear 3 times will become zero, so the result of this sum will be the single number.
def findSingleNumber(numbers: Array[Int]) = {
def add3(a: String, b: String): String = a.zipAll(b, '0', '0').map {
case (i, j) => ((i.toInt + j.toInt) % 3 + '0').toChar
}(collection.breakOut)
val numbersInBase3 = numbers.map(n => Integer.toString(n, 3).reverse)
Integer.parseInt(numbersInBase3.fold("0")(add3).reverse, 3)
}
scala> findSingleNumber(Array(10, 20, 30, 100, 20, 100, 10, 10, 20, 100))
res1: Int = 30
Or representing base 3 numbers as digit arrays:
def findSingleNumber(numbers: Array[Int]) = {
def toBase3(int: Int): Array[Int] =
Iterator.iterate(int)(_ / 3).takeWhile(_ != 0).map(_ % 3).toArray
def toBase10(arr: Array[Int]): Int =
arr.reverseIterator.foldLeft(0)(_ * 3 + _)
def add3(a: Array[Int], b: Array[Int]): Array[Int] = a.zipAll(b, 0, 0).map {
case (i, j) => (i + j) % 3
}
toBase10(numbers.map(toBase3).fold(Array.empty[Int])(add3))
}

Count number of 1 digits in 11 to the power of N

I came across an interesting problem:
How would you count the number of 1 digits in the representation of 11 to the power of N, 0<N<=1000.
Let d be the number of 1 digits
N=2 11^2 = 121 d=2
N=3 11^3 = 1331 d=2
Worst time complexity expected O(N^2)
The simple approach where you compute the number and count the number of 1 digits my getting the last digit and dividing by 10, does not work very well. 11^1000 is not even representable in any standard data type.
Powers of eleven can be stored as a string and calculated quite quickly that way, without a generalised arbitrary precision math package. All you need is multiply by ten and add.
For example, 111 is 11. To get the next power of 11 (112), you multiply by (10 + 1), which is effectively the number with a zero tacked the end, added to the number: 110 + 11 = 121.
Similarly, 113 can then be calculated as: 1210 + 121 = 1331.
And so on:
11^2 11^3 11^4 11^5 11^6
110 1210 13310 146410 1610510
+11 +121 +1331 +14641 +161051
--- ---- ----- ------ -------
121 1331 14641 161051 1771561
So that's how I'd approach, at least initially.
By way of example, here's a Python function to raise 11 to the n'th power, using the method described (I am aware that Python has support for arbitrary precision, keep in mind I'm just using it as a demonstration on how to do this an an algorithm, which is how the question was tagged):
def elevenToPowerOf(n):
# Anything to the zero is 1.
if n == 0: return "1"
# Otherwise, n <- n * 10 + n, once for each level of power.
num = "11"
while n > 1:
n = n - 1
# Make multiply by eleven easy.
ten = num + "0"
num = "0" + num
# Standard primary school algorithm for adding.
newnum = ""
carry = 0
for dgt in range(len(ten)-1,-1,-1):
res = int(ten[dgt]) + int(num[dgt]) + carry
carry = res // 10
res = res % 10
newnum = str(res) + newnum
if carry == 1:
newnum = "1" + newnum
# Prepare for next multiplication.
num = newnum
# There you go, 11^n as a string.
return num
And, for testing, a little program which works out those values for each power that you provide on the command line:
import sys
for idx in range(1,len(sys.argv)):
try:
power = int(sys.argv[idx])
except (e):
print("Invalid number [%s]" % (sys.argv[idx]))
sys.exit(1)
if power < 0:
print("Negative powers not allowed [%d]" % (power))
sys.exit(1)
number = elevenToPowerOf(power)
count = 0
for ch in number:
if ch == '1':
count += 1
print("11^%d is %s, has %d ones" % (power,number,count))
When you run that with:
time python3 prog.py 0 1 2 3 4 5 6 7 8 9 10 11 12 1000
you can see that it's both accurate (checked with bc) and fast (finished in about half a second):
11^0 is 1, has 1 ones
11^1 is 11, has 2 ones
11^2 is 121, has 2 ones
11^3 is 1331, has 2 ones
11^4 is 14641, has 2 ones
11^5 is 161051, has 3 ones
11^6 is 1771561, has 3 ones
11^7 is 19487171, has 3 ones
11^8 is 214358881, has 2 ones
11^9 is 2357947691, has 1 ones
11^10 is 25937424601, has 1 ones
11^11 is 285311670611, has 4 ones
11^12 is 3138428376721, has 2 ones
11^1000 is 2469932918005826334124088385085221477709733385238396234869182951830739390375433175367866116456946191973803561189036523363533798726571008961243792655536655282201820357872673322901148243453211756020067624545609411212063417307681204817377763465511222635167942816318177424600927358163388910854695041070577642045540560963004207926938348086979035423732739933235077042750354729095729602516751896320598857608367865475244863114521391548985943858154775884418927768284663678512441565517194156946312753546771163991252528017732162399536497445066348868438762510366191040118080751580689254476068034620047646422315123643119627205531371694188794408120267120500325775293645416335230014278578281272863450085145349124727476223298887655183167465713337723258182649072572861625150703747030550736347589416285606367521524529665763903537989935510874657420361426804068643262800901916285076966174176854351055183740078763891951775452021781225066361670593917001215032839838911476044840388663443684517735022039957481918726697789827894303408292584258328090724141496484460001, has 105 ones
real 0m0.609s
user 0m0.592s
sys 0m0.012s
That may not necessarily be O(n2) but it should be fast enough for your domain constraints.
Of course, given those constraints, you can make it O(1) by using a method I call pre-generation. Simply write a program to generate an array you can plug into your program which contains a suitable function. The following Python program does exactly that, for the powers of eleven from 1 to 100 inclusive:
def mulBy11(num):
# Same length to ease addition.
ten = num + '0'
num = '0' + num
# Standard primary school algorithm for adding.
result = ''
carry = 0
for idx in range(len(ten)-1, -1, -1):
digit = int(ten[idx]) + int(num[idx]) + carry
carry = digit // 10
digit = digit % 10
result = str(digit) + result
if carry == 1:
result = '1' + result
return result
num = '1'
print('int oneCountInPowerOf11(int n) {')
print(' static int numOnes[] = {-1', end='')
for power in range(1,101):
num = mulBy11(num)
count = sum(1 for ch in num if ch == '1')
print(',%d' % count, end='')
print('};')
print(' if ((n < 0) || (n > sizeof(numOnes) / sizeof(*numOnes)))')
print(' return -1;')
print(' return numOnes[n];')
print('}')
The code output by this script is:
int oneCountInPowerOf11(int n) {
static int numOnes[] = {-1,2,2,2,2,3,3,3,2,1,1,4,2,3,1,4,2,1,4,4,1,5,5,1,5,3,6,6,3,6,3,7,5,7,4,4,2,3,4,4,3,8,4,8,5,5,7,7,7,6,6,9,9,7,12,10,8,6,11,7,6,5,5,7,10,2,8,4,6,8,5,9,13,14,8,10,8,7,11,10,9,8,7,13,8,9,6,8,5,8,7,15,12,9,10,10,12,13,7,11,12};
if ((n < 0) || (n > sizeof(numOnes) / sizeof(*numOnes)))
return -1;
return numOnes[n];
}
which should be blindingly fast when plugged into a C program. On my system, the Python code itself (when you up the range to 1..1000) runs in about 0.6 seconds and the C code, when compiled, finds the number of ones in 111000 in 0.07 seconds.
Here's my concise solution.
def count1s(N):
# When 11^(N-1) = result, 11^(N) = (10+1) * result = 10*result + result
result = 1
for i in range(N):
result += 10*result
# Now count 1's
count = 0
for ch in str(result):
if ch == '1':
count += 1
return count
En c#:
private static void Main(string[] args)
{
var res = Elevento(1000);
var countOf1 = res.Select(x => int.Parse(x.ToString())).Count(s => s == 1);
Console.WriteLine(countOf1);
}
private static string Elevento(int n)
{
if (n == 0) return "1";
//Otherwise, n <- n * 10 + n, once for each level of power.
var num = "11";
while (n > 1)
{
n--;
// Make multiply by eleven easy.
var ten = num + "0";
num = "0" + num;
//Standard primary school algorithm for adding.
var newnum = "";
var carry = 0;
foreach (var dgt in Enumerable.Range(0, ten.Length).Reverse())
{
var res = int.Parse(ten[dgt].ToString()) + int.Parse(num[dgt].ToString()) + carry;
carry = res/10;
res = res%10;
newnum = res + newnum;
}
if (carry == 1)
newnum = "1" + newnum;
// Prepare for next multiplication.
num = newnum;
}
//There you go, 11^n as a string.
return num;
}

Find all possible combinations of a String representation of a number

Given a mapping:
A: 1
B: 2
C: 3
...
...
...
Z: 26
Find all possible ways a number can be represented. E.g. For an input: "121", we can represent it as:
ABA [using: 1 2 1]
LA [using: 12 1]
AU [using: 1 21]
I tried thinking about using some sort of a dynamic programming approach, but I am not sure how to proceed. I was asked this question in a technical interview.
Here is a solution I could think of, please let me know if this looks good:
A[i]: Total number of ways to represent the sub-array number[0..i-1] using the integer to alphabet mapping.
Solution [am I missing something?]:
A[0] = 1 // there is only 1 way to represent the subarray consisting of only 1 number
for(i = 1:A.size):
A[i] = A[i-1]
if(input[i-1]*10 + input[i] < 26):
A[i] += 1
end
end
print A[A.size-1]
To just get the count, the dynamic programming approach is pretty straight-forward:
A[0] = 1
for i = 1:n
A[i] = 0
if input[i-1] > 0 // avoid 0
A[i] += A[i-1];
if i > 1 && // avoid index-out-of-bounds on i = 1
10 <= (10*input[i-2] + input[i-1]) <= 26 // check that number is 10-26
A[i] += A[i-2];
If you instead want to list all representations, dynamic programming isn't particularly well-suited for this, you're better off with a simple recursive algorithm.
First off, we need to find an intuitive way to enumerate all the possibilities. My simple construction, is given below.
let us assume a simple way to represent your integer in string format.
a1 a2 a3 a4 ....an, for instance in 121 a1 -> 1 a2 -> 2, a3 -> 1
Now,
We need to find out number of possibilities of placing a + sign in between two characters. + is to mean characters concatenation here.
a1 - a2 - a3 - .... - an, - shows the places where '+' can be placed. So, number of positions is n - 1, where n is the string length.
Assume a position may or may not have a + symbol shall be represented as a bit.
So, this boils down to how many different bit strings are possible with the length of n-1, which is clearly 2^(n-1). Now in order to enumerate the possibilities go through every bit string and place right + signs in respective positions to get every representations,
For your example, 121
Four bit strings are possible 00 01 10 11
1 2 1
1 2 + 1
1 + 2 1
1 + 2 + 1
And if you see a character followed by a +, just add the next char with the current one and do it sequentially to get the representation,
x + y z a + b + c d
would be (x+y) z (a+b+c) d
Hope it helps.
And you will have to take care of edge cases where the size of some integer > 26, of course.
I think, recursive traverse through all possible combinations would do just fine:
mapping = {"1":"A", "2":"B", "3":"C", "4":"D", "5":"E", "6":"F", "7":"G",
"8":"H", "9":"I", "10":"J",
"11":"K", "12":"L", "13":"M", "14":"N", "15":"O", "16":"P",
"17":"Q", "18":"R", "19":"S", "20":"T", "21":"U", "22":"V", "23":"W",
"24":"A", "25":"Y", "26":"Z"}
def represent(A, B):
if A == B == '':
return [""]
ret = []
if A in mapping:
ret += [mapping[A] + r for r in represent(B, '')]
if len(A) > 1:
ret += represent(A[:-1], A[-1]+B)
return ret
print represent("121", "")
Assuming you only need to count the number of combinations.
Assuming 0 followed by an integer in [1,9] is not a valid concatenation, then a brute-force strategy would be:
Count(s,n)
x=0
if (s[n-1] is valid)
x=Count(s,n-1)
y=0
if (s[n-2] concat s[n-1] is valid)
y=Count(s,n-2)
return x+y
A better strategy would be to use divide-and-conquer:
Count(s,start,n)
if (len is even)
{
//split s into equal left and right part, total count is left count multiply right count
x=Count(s,start,n/2) + Count(s,start+n/2,n/2);
y=0;
if (s[start+len/2-1] concat s[start+len/2] is valid)
{
//if middle two charaters concatenation is valid
//count left of the middle two characters
//count right of the middle two characters
//multiply the two counts and add to existing count
y=Count(s,start,len/2-1)*Count(s,start+len/2+1,len/2-1);
}
return x+y;
}
else
{
//there are three cases here:
//case 1: if middle character is valid,
//then count everything to the left of the middle character,
//count everything to the right of the middle character,
//multiply the two, assign to x
x=...
//case 2: if middle character concatenates the one to the left is valid,
//then count everything to the left of these two characters
//count everything to the right of these two characters
//multiply the two, assign to y
y=...
//case 3: if middle character concatenates the one to the right is valid,
//then count everything to the left of these two characters
//count everything to the right of these two characters
//multiply the two, assign to z
z=...
return x+y+z;
}
The brute-force solution has time complexity of T(n)=T(n-1)+T(n-2)+O(1) which is exponential.
The divide-and-conquer solution has time complexity of T(n)=3T(n/2)+O(1) which is O(n**lg3).
Hope this is correct.
Something like this?
Haskell code:
import qualified Data.Map as M
import Data.Maybe (fromJust)
combs str = f str [] where
charMap = M.fromList $ zip (map show [1..]) ['A'..'Z']
f [] result = [reverse result]
f (x:xs) result
| null xs =
case M.lookup [x] charMap of
Nothing -> ["The character " ++ [x] ++ " is not in the map."]
Just a -> [reverse $ a:result]
| otherwise =
case M.lookup [x,head xs] charMap of
Just a -> f (tail xs) (a:result)
++ (f xs ((fromJust $ M.lookup [x] charMap):result))
Nothing -> case M.lookup [x] charMap of
Nothing -> ["The character " ++ [x]
++ " is not in the map."]
Just a -> f xs (a:result)
Output:
*Main> combs "121"
["LA","AU","ABA"]
Here is the solution based on my discussion here:
private static int decoder2(int[] input) {
int[] A = new int[input.length + 1];
A[0] = 1;
for(int i=1; i<input.length+1; i++) {
A[i] = 0;
if(input[i-1] > 0) {
A[i] += A[i-1];
}
if (i > 1 && (10*input[i-2] + input[i-1]) <= 26) {
A[i] += A[i-2];
}
System.out.println(A[i]);
}
return A[input.length];
}
Just us breadth-first search.
for instance 121
Start from the first integer,
consider 1 integer character first, map 1 to a, leave 21
then 2 integer character map 12 to L leave 1.
This problem can be done in o(fib(n+2)) time with a standard DP algorithm.
We have exactly n sub problems and button up we can solve each problem with size i in o(fib(i)) time.
Summing the series gives fib (n+2).
If you consider the question carefully you see that it is a Fibonacci series.
I took a standard Fibonacci code and just changed it to fit our conditions.
The space is obviously bound to the size of all solutions o(fib(n)).
Consider this pseudo code:
Map<Integer, String> mapping = new HashMap<Integer, String>();
List<String > iterative_fib_sequence(string input) {
int length = input.length;
if (length <= 1)
{
if (length==0)
{
return "";
}
else//input is a-j
{
return mapping.get(input);
}
}
List<String> b = new List<String>();
List<String> a = new List<String>(mapping.get(input.substring(0,0));
List<String> c = new List<String>();
for (int i = 1; i < length; ++i)
{
int dig2Prefix = input.substring(i-1, i); //Get a letter with 2 digit (k-z)
if (mapping.contains(dig2Prefix))
{
String word2Prefix = mapping.get(dig2Prefix);
foreach (String s in b)
{
c.Add(s.append(word2Prefix));
}
}
int dig1Prefix = input.substring(i, i); //Get a letter with 1 digit (a-j)
String word1Prefix = mapping.get(dig1Prefix);
foreach (String s in a)
{
c.Add(s.append(word1Prefix));
}
b = a;
a = c;
c = new List<String>();
}
return a;
}
old question but adding an answer so that one can find help
It took me some time to understand the solution to this problem – I refer accepted answer and #Karthikeyan's answer and the solution from geeksforgeeks and written my own code as below:
To understand my code first understand below examples:
we know, decodings([1, 2]) are "AB" or "L" and so decoding_counts([1, 2]) == 2
And, decodings([1, 2, 1]) are "ABA", "AU", "LA" and so decoding_counts([1, 2, 1]) == 3
using the above two examples let's evaluate decodings([1, 2, 1, 4]):
case:- "taking next digit as single digit"
taking 4 as single digit to decode to letter 'D', we get decodings([1, 2, 1, 4]) == decoding_counts([1, 2, 1]) because [1, 2, 1, 4] will be decode as "ABAD", "AUD", "LAD"
case:- "combining next digit with the previous digit"
combining 4 with previous 1 as 14 as a single to decode to letter N, we get decodings([1, 2, 1, 4]) == decoding_counts([1, 2]) because [1, 2, 1, 4] will be decode as "ABN" or "LN"
Below is my Python code, read comments
def decoding_counts(digits):
# defininig count as, counts[i] -> decoding_counts(digits[: i+1])
counts = [0] * len(digits)
counts[0] = 1
for i in xrange(1, len(digits)):
# case:- "taking next digit as single digit"
if digits[i] != 0: # `0` do not have mapping to any letter
counts[i] = counts[i -1]
# case:- "combining next digit with the previous digit"
combine = 10 * digits[i - 1] + digits[i]
if 10 <= combine <= 26: # two digits mappings
counts[i] += (1 if i < 2 else counts[i-2])
return counts[-1]
for digits in "13", "121", "1214", "1234121":
print digits, "-->", decoding_counts(map(int, digits))
outputs:
13 --> 2
121 --> 3
1214 --> 5
1234121 --> 9
note: I assumed that input digits do not start with 0 and only consists of 0-9 and have a sufficent length
For Swift, this is what I came up with. Basically, I converted the string into an array and goes through it, adding a space into different positions of this array, then appending them to another array for the second part, which should be easy after this is done.
//test case
let input = [1,2,2,1]
func combination(_ input: String) {
var arr = Array(input)
var possible = [String]()
//... means inclusive range
for i in 2...arr.count {
var temp = arr
//basically goes through it backwards so
// adding the space doesn't mess up the index
for j in (1..<i).reversed() {
temp.insert(" ", at: j)
possible.append(String(temp))
}
}
print(possible)
}
combination(input)
//prints:
//["1 221", "12 21", "1 2 21", "122 1", "12 2 1", "1 2 2 1"]
def stringCombinations(digits, i=0, s=''):
if i == len(digits):
print(s)
return
alphabet = 'ABCDEFGHIJKLMNOPQRSTUVWXYZ'
total = 0
for j in range(i, min(i + 1, len(digits) - 1) + 1):
total = (total * 10) + digits[j]
if 0 < total <= 26:
stringCombinations(digits, j + 1, s + alphabet[total - 1])
if __name__ == '__main__':
digits = list()
n = input()
n.split()
d = list(n)
for i in d:
i = int(i)
digits.append(i)
print(digits)
stringCombinations(digits)

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